No one can teach mathematics without knowing mathematics but not everyone who knows mathematics can teach it well. Below are two tasks about teaching fractions and decimals that would give us a sense of the kind of mathematical knowledge we teachers need to know apart from knowledge of the content of mathematics. As teachers it is expected of us to have knowledge of students difficulties and misconceptions in specific domains of mathematics. We are also expected to know the different representations or models of concepts to design an effective instruction. The two tasks were used in a study about mathematical knowledge for teaching of pre-service teachers.
Task 1
You are teaching in 7th grade. You want to work on multiplication of fractions, using the following numbers:
a) 10 x 3 b) 10 x 3/4 c. 10 x 1 1/5 d. 10/11 x 1 1/5
Create a problem using an everyday context, accessible to students and easily visualized, that uses the repeated addition sense for multiplication;
Prepare an illustration that works and that you could use for all numbers to help students visualize the operation;
Show, for each case, with the illustration and specific explanations, how one can make sense of c) from the answer obtained in a).
Task 2
Arrange the following numbers from the least to the greatest:
2.46 2.254 2.3 2.052 2.32
Many of your students have written:
2.052 2.3 2.32 2.46 2.254
An others have written:
2.052 2.254 2.32 2.46 2.3
Complete the following steps:
Describe and make sense of the error(s) committed by students;
Find a similar task in which the students’ reasoning would lead to the same error, confirming their strategy;
Find a similar task in which the students’ reasoning would lead to a right answer;
You may use the comment section below to answer the questions or share your thoughts about mathematics teaching. I hope you find time to discuss this with your co-teachers.
Negative numbers, the irrationals, and imaginary numbers are not that easy to make sense of for many students. But this is something understandable. One only needs to check-out the histories of these numbers. The mathematicians themselves took a long time to accept and make sense of them. But fractions? How can something so natural, useful, and so much a part of our everyday life be so difficult? Didn’t we learn what’s half before we even learn to count to 10? I’m sure this was true even with our brother cavemen. So how come the sight of a fraction enough to scare the wits out of many of our pupils and yes, adults, too?
Fractions are used to represent seemingly unrelated mathematical concepts and this is what makes these numbers not easy to make sense of and work with. In mathematics, fractions are used to represent a:
Part-whole relationship – the fraction 2/3 represents a part of a whole, two parts of three equal parts;
Quotient – 2/3 means 2 divided by 3;
Ratio – as in two parts to three parts; and
Measure – as in measure of position, e.g, 2/3 represents the position of a point on a number line.
Of these four, it is the part-whole relationship that dominates textbooks. For many this conception is what they all know about fractions. While it is also the easiest of the four to make sense of, students requires series of learning activities to fully understand part-whole relationship . Crucial to this notion is the ability to partition a continuous quantity or a set of discrete objects into equal sized parts. Below are sample tasks to teach/assess this understanding. They call for visualizing skills.
Of course understanding fractions involve more than just being able to use them in representing quantities in different contexts. There’s the notion of fraction equivalence, which is one of the most important mathematical ideas in the primary school mathematics and a major difficulty. This difficulty is ascribed to the multiplicative nature of this concept. There’s the notion of comparison of fraction which includes finding the order relation between two fractions. And if your students are having a hard time on comparing fractions you can check their understanding of equivalence of two fractions. It could be the culprit. And let’s not forget the operations on fractions. An understanding of the procedure for adding, subtracting, multiplying, and dividing of fractions depends on students’ depth of understanding of the different ways fractions are conceived, on the way fractions are used to represent quantities, on the idea of equivalent fractions, and on order relation between fractions, and many others such as the meaning of the operations themselves.
A study has been conducted categorizing students levels of conception of fractions, at least up to addition operation. Just click on the link to read the summary.
It is misleading to put fractions alongside the sets of numbers – counting, whole, integers, rational, irrational and real. The diagram below which are in many Mathematics I (Year 7) textbooks is inviting misconceptions.
WRONG WAY
Fraction is a form for writing numbers just like the decimals, percents, and other notations that use exponents and radicals, etc.
The fraction form of numbers is used to describe quantities that is 1) part of a whole, 2) part of a set, 3) ratio, and 4) as an indicated operation. Yes, it can also represent all the rational numbers but it doesn’t make fractions another kind of number or as another way of describing the rational numbers. Decimals can represent both the rational and the irrational numbers (approximately) but it is not a separate set of numbers or used as another way of describing the real numbers! Note that I’m using the word number not in everyday sense but in mathematical sense. In Year 7, where learners are slowly introduced to the rigor of mathematics and to the real number system, I suggest you start calling the numbers in its proper name.
I prefer the Venn diagram to show the relationships among the different kinds of numbers like the one shown below:
The Real Numbers
The diagram shows that the set of real numbers is composed of the rational and the irrational numbers. The integers are part of the set of rational numbers just like the counting numbers are members of the set of whole numbers and the whole numbers are members of the set of integers. The properties of each of these set of numbers can be investigated. We do not investigate if fraction is closed or is commutative under a certain operation for example, but we do it for the rational numbers.
The post It Ain’t No Repeated Addition by Devlin launched a math war over the definition of multiplication. Here’s an excerpt from that post:
“Multiplication simply is not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not. Multiplication of natural numbers certainly gives the same result as repeated addition, but that does not make it the same. Riding my bicycle gets me to my office in about the same time as taking my car, but the two processes are very different. Telling students falsehoods on the assumption that they can be corrected later is rarely a good idea. And telling them that multiplication is repeated addition definitely requires undoing later.”
What is multiplication?
Multiplication is repeated addition is definitely not correct. Counterexample: 1/2 x 1/4. Try also doing it with integers like -5 x -4 (not that you need two counterexamples to reject a statement). But is it correct to say that in the set of whole numbers multiplication is repeated addition? I think not. You can get the result by repeated addition for this set of number yes, but that does not make repeated addition a definition of multiplication. An operation is not defined by the strategy of getting its result.
But, should teachers in the grades stop telling pupils that multiplication is repeated addition? YES! In fact, they should refrain from telling pupils any rule at all. The pupils are perfectly capable of figuring things like these by themselves given the right task/activity and good facilitation by the teacher.
And let us suppose that students get this conception that multiplication is repeated addition, is there really a problem? Their world revolve around whole numbers so it’s only logical that this will be their understanding of it. Generalizing is a natural human tendency. Something must be wrong if they will not make this connection between multiplication and addition.
What is wrong with “undoing” later? Mathematics is man-made and there’s also a lot of trial and error part in its development. That is why “undoing” and rejection by counterexample are legitimate processes . And, isn’t ‘undoing’ part of teaching? Good teachers are those who can find out or know what they should be ‘undoing’ when they teach mathematics. ‘Multiplication is repeated addition’ is only one of many ‘over-generalizations’ pupils will make that teachers need to carefully undo later. There’s “when you multiply, you make it bigger”, or “the sum of two numbers is always bigger than any of the two you added”, etc. One way to prevent an over-generalization is to offer a counterexample. But where will you get that counterexample when their math still revolves around the world of whole numbers!
As teachers, don’t we all love that part of teaching where we challenge students’ assumptions? I’m not saying that we should deliberately lead pupils to over-generalizations so we have something to undo later. For example, we don’t lead them to “division is repeated subtraction”? Most of the time oversimplifying mathematics is not a good idea.
